Solving Quadratic Equations: Foundations

[Video: Introduction to Quadratic Equations]

Learn multiple methods for solving quadratic equations, a fundamental algebra skill tested frequently on the SAT.

What are Quadratic Equations?

Equations of the form ax² + bx + c = 0, where a ≠ 0.

Key Features

Graph is a parabola
Maximum of two real solutions (roots)
Solutions represent x-intercepts of the parabola

Solving Methods

1. Factoring

When ax² + bx + c can be factored into (dx + e)(fx + g) = 0

Steps:

Set equation equal to zero
Factor completely
Set each factor equal to zero
Solve the resulting linear equations

Example: Solve x² - 5x + 6 = 0

1. Factor: (x - 2)(x - 3) = 0

2. Solutions: x - 2 = 0 → x = 2

x - 3 = 0 → x = 3

2. Quadratic Formula

Works for any quadratic equation:

x = [-b ± √(b² - 4ac)]/(2a)

Example: Solve 2x² + 3x - 5 = 0

a = 2, b = 3, c = -5

x = [-3 ± √(9 - 4(2)(-5))]/4

= [-3 ± √49]/4

= (-3 ± 7)/4

Solutions: x = 1 and x = -2.5

3. Completing the Square

Useful when quadratic formula is difficult to apply:

Move constant term to other side
Divide by coefficient of x² if needed
Add (b/2)² to both sides
Write left side as perfect square
Take square root of both sides
Solve for x

Example: Solve x² + 6x - 7 = 0

1. x² + 6x = 7

2. Add (6/2)² = 9: x² + 6x + 9 = 16

3. (x + 3)² = 16

4. x + 3 = ±4

5. x = -3 ± 4 → x = 1 or x = -7

Key Takeaways

Try factoring first if the equation looks factorable
Quadratic formula works for all cases but requires careful calculation
Completing the square is useful for vertex form conversion
Check solutions by plugging them back into the original equation
Remember that some quadratics have no real solutions (when discriminant is negative)

Practice Question

What are the solutions to the equation 3x² - 7x - 6 = 0?

A) x = -2/3 and x = 3

B) x = -3/2 and x = 2

C) x = -1 and x = 6

D) x = -1/3 and x = 6

Consider these approaches:

Try factoring first: look for two numbers that multiply to 3×(-6)=-18 and add to -7
Alternatively, use the quadratic formula with a=3, b=-7, c=-6
Check your solutions by plugging them back into the original equation

Solution using factoring:

1. Find two numbers that multiply to -18 and add to -7: -9 and 2

2. Rewrite middle term: 3x² - 9x + 2x - 6 = 0

3. Factor by grouping: (3x² - 9x) + (2x - 6) = 0

3x(x - 3) + 2(x - 3) = 0

(3x + 2)(x - 3) = 0

4. Solutions: 3x + 2 = 0 → x = -2/3

x - 3 = 0 → x = 3

The correct answer is A) x = -2/3 and x = 3.